12,1

b(n) for n >= 2 may be defined as follows. For a full n-dimensional shift, let M(N)=\sum_{L} O(L)/exp(h[L])

where the sum is over subgroups L of finite index in Z^n, O(L) is the number of points with stabilizer L and exp(h) is the number of symbols.

Then M(N) is asymptotic to a rational times a power of \pi times a product of values of the zeta function at odd integers and b(n) is the numerator of that rational.

R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc. 137 (2009), 1499-1507.

M(N)=residue((\zeta(z+1)...\zeta(z-n+2)N^z)/z,z=n-1)=(a(n)/b(n))N^{d-1}\pi^{\lfloor\frac{n}{2}\rfloor(\lfloor\frac{n}{2}\rfloor+1)}\prod_{j=1}^{\lfloor(n-1)/2\rfloor}\zeta(2j+1)

For n=12, using the formula in terms of residues, we have residue(zeta(z+1)...zeta(z-10)N^z/z,z=11)=(691/3168740859543387253125000)zeta(3)zeta(5)zeta(7)zeta(9)zeta(11)\pi^42N^11, so b(12)=691.

residue(product(Zeta(z-j), j=-1..(n-2))*N^z/z, z=n-1) generates an expression from which b(n) can be read off

This is the numerator of a rational sequence whose denominator is A159282

Sequence in context: A141590 A046988 A029825 this_sequence A106281 A127341 A135316

Adjacent sequences: A159280 A159281 A159282 this_sequence A159284 A159285 A159286

easy,frac,nonn

Thomas Ward (t.ward(AT)uea.ac.uk), Apr 08 2009